Question

Derive the Boolean expression of a combination logic from the following truth table, where A, B, C are input variables and D is output. Draw the circuit diagram to implement it. Show your working steps. The full subtractor is a combinational circuit, which is used to perform subtraction of three input bits: the minuend X, subtrahend Y, and borrow in B_in. The full subtractor generates two outputs bits: the difference D and borrow out B_out. B_in is set when the previous digit borrowed from X. Thus B_in is also subtracted from X as well as the subtrahend Y. Or in symbols: X-Y- B_in. The full subtractor generates a borrow out when it needs to borrow from the next digit. Since we are subtracting X by Y and B_in, a borrow out needs to be generated when X < Y < + B_in. When borrow out is generated, 2 is added in the current digit (This is similar to the subtraction algorithm in decimal. Instead of adding 2, we add 10 when we borrow). Therefore, the arithmetic equation for the full subtractor can be expressed as: D = X-Y-B_in + 2B_out B_out = 1, if X < (Y + B_in) For example: If inputs X = 0, Y = 0, B_in = 1, the output B_out = 1, D = 1. If inputs X = 0, Y = 1, B_in = 1, the output B_out = 1, D = 0. 1) List the true table for the full subtractor: 2) Derive the logic equation for the full subtractor: 3) Draw the circuit diagram for the full subtractor (use XOR gate, Exclusive NOR gate).

Answer 1

a)Boolean expression and circuit diagram is given below:

A B C ABC ABC 'also be smpǐ수ed as D car

b)1)Truth table for full subtractor :

TRUTH TABLE FOR FULL SUBTRACTOR Input Output 囗

2)

Map simplification D BBin D = AB Bint ABB:n can be simplified further as Bin (Ae) i(s)

D=B_in$\tiny \bigoplus$(A$\tiny \bigoplus$B) and B_out =$\large {\bar{A}$B_in+ $\large {\bar{A}$ B + BB_in

3)Full subtractor using XOR gate is given below

Boolom unchon circuit -for full Subtractor 3 Bin 8 Bout 8i